Prove that root over 5 is an irational number.
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hello there !!!
remember two numbers having no common factor other than one are said to be co prime :)
now
if possible, let √5 is a rational then there exist positive co prime a and b such that
√5=a/b
=>√5b=a
squaring both the side , we get
=> 5b²=a²-------(1)
=> a² is divisible by 5 [since 5b² is divisible by 5 and 5b²=a²]
5 is a prime which divides a² so it will divide a also
now let, a=5c for some integer c
putting it in (1)
we get
5b²=25c²
=>b²=5c²
=> b² is divisible by 5 [since 5c² is divisible by 5 and 5c²=b²]
5 is a prime which divides b² so it will divide b also
but this contradicts the fact that a and b are co prime
hence , √5 is an irrational number
hope this helped you
remember two numbers having no common factor other than one are said to be co prime :)
now
if possible, let √5 is a rational then there exist positive co prime a and b such that
√5=a/b
=>√5b=a
squaring both the side , we get
=> 5b²=a²-------(1)
=> a² is divisible by 5 [since 5b² is divisible by 5 and 5b²=a²]
5 is a prime which divides a² so it will divide a also
now let, a=5c for some integer c
putting it in (1)
we get
5b²=25c²
=>b²=5c²
=> b² is divisible by 5 [since 5c² is divisible by 5 and 5c²=b²]
5 is a prime which divides b² so it will divide b also
but this contradicts the fact that a and b are co prime
hence , √5 is an irrational number
hope this helped you
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